EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 2720-2725 ISSN 1307-5543 – ejpam.com Published by New York Business Global On the Symmetric Block Design With Parameters (220, 73, 24) Admitting a Group of Order 73 Menderes Gashi Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Prishtina, Avenue ”Mother Teresa” 5, 10000 Prishtina, Kosovo Abstract. In this paper, we have demonstrated that for a putative symmetric block design D with parameters (220,73,24) constructed by group G of order 73, there exists only one orbit structure up to isomorphism. The full automorphism group for this orbit structure is provided. 2020 Mathematics Subject Classifications: 05B05 Key Words and Phrases: Symmetric block design, Orbit structure, Automorphism group 1. Introduction and preliminaries A 2−(v, k, λ) block design (P,B, I) is said to be symmetric if the relation |P| = |B| = v holds and in that case we often speak of a symmetric design with parameters (v, k, λ). The integer n = k − λ is called the order of the symmetric block design. The collection of the parameter sets (v, k, λ) for which a symmetric 2 − (v, k, λ) block design exists is often called the ”spectrum”. The determination of the spectrum for symmetric block designs is a widely open problem. For example, a finite projective plane of order n is a symmetric design with parameters (n2+n+1, n+1, 1) and it is still unknown whether finite projective planes of non–prime–power order may exist at all. The existence/non-existence of a symmetric block design has often required ”ad hoc” treatments even for a single parameter set (v, k, λ). The most famous instance of this circumstance is perhaps the non-existence of the projective plane of order 10, see [9]. It is worthwhile to investigate symmetric block designs with supplementary character- istics, frequently entailing the premise that a non-trivial automorphism group acts on the design under scrutiny. See for instance [5]. Investigating symmetric block designs of order 49 within the realm of symmetric block designs of square order holds notable significance. Despite the existence of 15 potential parameters (v, k, λ) for symmetric block designs of order 49, only limited results have been established thus far (see [4], [6]). Given the large number of points (blocks) in symmetric DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5377 Email address: menderes.gashi@uni-pr.edu (M. Gashi) https://www.ejpam.com 2720 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) M. Gashi / Eur. J. Pure Appl. Math, 17 (4) (2024), 2720-2725 2721 block designs of order 49, investigating sporadic cases can be highly challenging, unless one assumes the existence of a collineation group. Several techniques exist for constructing symmetric block designs, each demonstrating effectiveness in specific situations. In this instance, we opt for the tactical decompositions method, as employed by Z. Janko in [7], also referenced in [8], assuming the action of a particular automorphism group on the design under construction. This paper focuses on analyzing a symmetric block design denoted as D = (P,B, I) with parameters (220, 73, 24). At present, the existence or non-existence of such a design remains uncertain to the best of our knowledge. Additionally, we posit that the specified design admits a particular automorphism group of order 73. We expect that the reader has a grasp of fundamental concepts in design theory, as outlined in references such as [2], [3] and [10]. If g denotes an automorphism of a symmetric design D characterized by parameters (v, k, λ), it is observed that g fixes an equal number of points and blocks, as detailed in [10, Theorem 3.1, p.78]. The sets of these fixed elements we denote by FP(g) and FB(g) each, and their number simply by |F (g)|. For number of fixed points shall we use the following upper bound, as delineated in [10, Corollary 3.7, p. 82]: |F (g)| ≤ k + √ k − λ. (1) It’s established that an automorphism group G of a symmetric block design exhibits an equal number of orbits on both the set of points P and the set of blocks B, as outlined in [10, Theorem 3.3, p.79]. This number is denoted by t. We utilize the notation and terminology introduced in Section 1 of [5], recapitulating certain essential relations for the reader’s convenience. Consider D as a symmetric block design characterized by parameters (v, k, λ), with G representing a subgroup of the auto- morphism group Aut(D). The point orbits of G on P are denoted as P1,P2, . . .Pt, and the line orbits of G on B as B1,B2, . . .Bt. Let |Pr| = ωr and |Bi| = Ωi. Clearly, t∑ r=1 ωr = t∑ i=1 Ωi = v. (2) Consider γir as the number of points from Pr situated on a block from Bi; evidently, this number remains invariant regardless of the chosen block. Similarly, let Γjs be the number of blocks from Bj intersecting a point from Ps. It is evident that, t∑ r=1 γir = k and t∑ j=1 Γjs = k. (3) By [3, Lemma 5.3.1. p.221], the division of both the point set P and the block set B constitutes a tactical decomposition of design D as defined in [3, p.210]. Consequently, the ensuing equations are valid: Ωi · γir = ωr · Γir, (4) M. Gashi / Eur. J. Pure Appl. Math, 17 (4) (2024), 2720-2725 2722 t∑ r=1 γirΓjr = λΩj + δij(k − λ), (5) t∑ i=1 Γirγis = λωs + δrs(k − λ), (6) where δij , δrs are the Kronecker symbols. For verification of these equations, readers are encouraged to consult [3] and [5]. Com- bining Equation (5) with (4) results in t∑ r=1 Ωj ωr γirγjr = λΩj + δij(k − λ). (7) Definition 1. We denote [Li, Lj ] = t∑ r=1 Ωj ωr γirγjr, 1 ≤ i, j ≤ t and term these expressions as the orbit products. The (t× t)-matrix (γir) is called the orbit structure of the block design D. An automorphism of an orbit structure entails a permutation of rows followed by a permutation of columns, maintaining the matrix unchanged. It’s evident that the collec- tion of all such automorphisms forms a group, referred to as the automorphism group of that orbit structure. The initial step in constructing a design is to identify all potential orbit structures. The subsequent step, typically referred to as indexing, involves specifying which γir points of the orbit Pr lie on the blocks of the block orbit Bi for each coefficient γir of the orbit matrix. Naturally, this process only needs to be performed for a representative of each block orbit, since the other blocks in that orbit can be generated by producing all G-images of the selected representative. 2. Main results Let D represent the symmetric block design with parameters (220, 73, 24). Given that v = 1 + 3 · 73, to construct the symmetric block design D, we employ the cyclic group G = ⟨ρ|ρ73 = 1⟩ of order 73 as a collineation group. Lemma 1. Let ρ be an element of G with o(ρ) = 73. Then ⟨ρ⟩ fixes exactly one point and one block. Proof. According to [10, Theorem 3.1], the group ⟨ρ⟩ fixes the same number of points and blocks. Let this number be denoted by f . Clearly, f ≡ 220(mod 73), which means f ≡ 1(mod 73). The upper bound (1) for the number of fixed points gives f ∈ {1, 74}. M. Gashi / Eur. J. Pure Appl. Math, 17 (4) (2024), 2720-2725 2723 Since o(ρ) > λ, applying a result from M. Aschbacher [1, Lemma 2.6, p.274] necessitates the fixed structure being a subdesign of D. However, there is no symmetric block design with v = 74 and λ = 24 (no k ∈ IN satisfies 24 · (v − 1) = k · (k − 1)). Therefore, f must be equal to 1. We set PI = {I0, I1, · · · , I72}, I = 1, 2, 3, for the non-trivial point orbits of the group G. Consecuently, G uniquely acts as a permutation group on these point orbits. Therefore, the generator of G can be defined as follows. ρ = (∞)(I0, I1, · · · , I72), I = 1, 2, 3, where ∞ is the fixed point of the collineation, the non-trivial ⟨ρ⟩-orbits are the numbers 1, 2, 3, and ∞, 10, 11, · · · , 372 represent all points of the symmetric block design D. In the following steps, we will construct a representative block for each block orbit. The ⟨ρ⟩−fixed block can be expressed as: L1 = (1011 · · · 172) or L1 = 173. Let L2, L3, and L4 be the representative blocks for the three non-trivial block orbits. The second ρ–orbit block, L2, of block design D, constructed by the collineation ρ, can be expressed as L2 = ∞1a12a23a3 , where ai, i = 1, 2, 3, represent the multiplicities of the occurrence of orbit numbers 1, 2, and 3 in the orbit block L2. The multiplicities of these orbit numbers must satisfy the following conditions: a1 + a2 + a3 = 72. Because |L1 ∩ L2| = 24, we have a1 = 24. From (7) we have [L2, L2] = 73/1 · 1 · 1+73/73 ·a21+73/73 ·a22+73/73 ·a23 = 24 · 73+73− 24 = 1801, i.e. a21 + a22 + a23 = 1728 or a22 + a23 = 1152, whence it follows that the multiplicities of appearances in block L2 yield the constraints 0 ≤ ai ≤ 33, i = 2, 3. To minimize isomorphic cases in the orbit structures at the final stage, we can assume without loss of generality that a2 ≥ a3 for block L2. Using computational methods, we have demonstrated that there exists exactly one orbit type for block L2 that meets the aforementioned conditions: a1 a2 a3 1. 24 24 24 M. Gashi / Eur. J. Pure Appl. Math, 17 (4) (2024), 2720-2725 2724 The form of the third orbit block L3, created using the collineation ρ, is as follows: L3 = 1b12b23b3 . Here, bi, where i = 1, 2, 3 represent the frequencies of orbit numbers 1, 2, and 3 appearing in orbit block L3. The occurrences of orbit numbers must adhere to the following criteria: b1 + b2 + b3 = 73. [L1 ∩ L3] = 24 implies b1 = 24. From (7) we have [L3, L3] = b21 + b22 + b23 = 24 · 73 + 73− 24 = 1801 or b22 + b23 = 1225. Based on the previous relation, we deduce the constraints 0 ≤ bi ≤ 35, where i = 2, 3. [L2, L3] = a1b1 + a2b2 + a3b3 = 24 · 73 = 1752. Through computational analysis, we have confirmed the existence of precisely two orbit types for block L3 that meet the aforementioned criteria: b1 b2 b3 1. 24 28 21 2. 24 21 28 It is evident that among the contenders for block L3 are also blocks L4. Conse- quently, we examine pairs of blocks {L3, l4} that are mutually compatible. Through this approach, we have determined that, up to isomorphism, there exists precisely one orbit structure for the symmetric block design with parameters (220, 73, 24) under the action of the collineation ρ of order 73: Orbit structure: SO 1 73 73 73 0 73 0 0 1 24 24 24 0 24 28 21 0 24 21 28 Full automporphism group of the orbit stucture is: Aut(SO) = {1, (3 4)(3̄ 4̄)} Thus we have Theorem 1. There exists precisely one orbit structure, up to isomorphism, for the sym- metric block design D characterized by parameters (220, 73, 24) and accommodating the group G of order 73. Remark 1. The specific indexing of this orbit structure to generate an example remains an unresolved issue. REFERENCES 2725 Acknowledgements I would like to express my gratitude to the anonymous referees for their helpful com- ments and suggestions that have improved the quality of the paper. References [1] M. Aschbacher. On collineation groups of symmetric block designs. J. Comb. Theory Ser. A, 11:272–281, 1971. 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